Note that things get even more complex once we're outside of classical (Newtonian) dynamics -- that is to say, even simultaneity itself is relative to the reference frame of the observer.
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Joseph Weissman♦Feb 13 '13 at 0:25

"simultaneous causation" seems to be a contradiction in terms.
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DruxFeb 17 '13 at 12:00

5 Answers
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Special relativity rules out the possibility of absolute simultaneity. Read from the wikipedia page:

The theory of relativity postulates a maximum rate of transmission of information as the speed of light, and one consequence is that simultaneity at separated locations always is relative to the observer.

Einstein had his own views on how we should treat our ordinary concept of simultaneity in light of special relativity. It certainly can't be (if we speak truly) that we are using the notion of absolute simultaneity. He wrote in a early paper that whether two events in distinct spatial locations were simultaneous has no definite answer and should be settled by convention. Whether special relativity is compatible with a non-conventional analysis of simultaneity is a matter of debate (see here for more discussion).

Taking a look at the paper you cite, they are fairly upfront that they are assuming classical physics. While this might be bad form for a physicist, it makes good sense within the context of the debate they are engaged in. Their thesis is NOT "since time actually has such and such a structure this view of causation is correct". Rather, they're stating something conditional in form.

Also, it is worth noting that for most everyday events, the differences between Newtonian and Relativistic are not too important. Newtonian mechanics approximates its more accurate relativistic counterpart when dealing with slow moving (i.e., low speed compared to the speed of light) medium sized objects (see the section on "limits of validity"). Most of the instances of causation we, as philosophers, are interested in analyzing are these everyday occurrences where pretending that classical physics is correct won't steer us too wrong. Since it is a simpler theory that is easier to understand and draw conclusions from, the hope is that it is still accurate enough to provide material for an accurate analysis of (everyday) causation.

First of all in mathematical physics we are treating idealisations of the world out there. There has been a continuous debate from antiquity as to what is meant by a continuum - is it punctual, that is made from points, or is there more? In classical geometry as done by Euclid, he allows for points on lines but is silent as to whether lines are exactly all the points on the line. His geometry is synthetic. Descarte actually took that step (which paved the way for Calculus), his geometry is analytic.

Certainly the most common notion treats the continuum as made up of points, for example the calculus as envisaged by Newton & Leibniz does this. But already then Leibniz declared the neccesity for a analysis situs.

This was eventually formalised as topology. Over and above the idealisation of the continuum as a set of points one explicitly says how they cohere. It was also noticed that then one could actually do away with the notion of points altogether and just have the notion of cohesiveness (the theory of locales).

When you have a theory that has no points then in a precise sense it is meaningless to ask for exact simultaneity. But what this actually means is that one must expand what one can mean by this in this new context.

Whereas there has been a motion in Physics to atomising nature, from the atoms of matter, to quanta of energy and conjecturally spacetime; there has been an opposite motion in which the these atoms as point particles are seen to be problematic. Hence wave-particle duality (aka wavicles!), string and brane theory.

So far I've discussed simultaneity in the small which appears to be what you're asking about, but as a couple of the other posters have pointed out, there are problems with simultaneity in the large as elucidated by Einstein in his special theory. The laws of motion that we instinctively take for granted and formalised by Galileo are not correct - there is an absolute speed (the speed of light). Now speed as a formal concept ties together time and space, so how these are actually related must be modifed. This was Einsteins accomplishment. Whereas the older paradigm treated space & time essentially independent (so there are two notions of simultaneity - one for time & the other for space) he showed that they must be treated together (so there is a single notion of simultaneity using the idea of relativistic distance which combines information about both space and time).

It turns out we can keep the old idea of simultaneity when we are talking about events that are at the same location of space, but we cannot if they aren't, we must then use the relativistic notion.

Whitehead criticizes Einstein's belief in "absolute speed" as not consistent with a relativistic universe. We all know Einstein admitted himself it was his greatest error fudging this "cosmological constant." The most we can say about a LIMIT extends only to our "cosmic epoch." I would suggest a thorough review of Einstein in light of Whitehead's critique. His notion of space is algebraic, that of lived experience and organic. Not a product of dead geometries and recondite measurements. Whitehead discarded the uniformity of space in Einstein's GR for a more robust extensive continuum.
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Myron Moses JacksonFeb 17 '13 at 18:44

Whitehead argues in Part II of Process and Reality that to accept simultaneities in the universe would violate the Ontological Principle which is an inflexible or categoreal presupposition of his speculative metaphysics.

He writes: “According to the classical ‘uniquely serial’ view of time, two contemporary actual entities define the same actual world. According to the modern view no two actual entities define the same actual world. Actual entities are called ‘contemporary’ when neither belongs to the ‘given’ actual world de¬fined by the other. The differences between the actual worlds of a pair of contemporary entities, which are in a certain sense ‘neighbours,’ are negligible for most human purposes. Thus the difference between the ‘classical’ and the ‘relativity’ view of time only rarely has any important relevance. I shall always adopt the relativity view; for one reason, because it seems better to accord with the general philosophical doctrine of relativity which is presupposed in the philosophy of organism; and for another reason, because with rare exceptions the classical doctrine can be looked on as a special case of the relativity doctrine—a case which does not seem to accord with experimental evidence. In other words, the classical view seems to limit a general philosophical doctrine; it is the larger assumption; and its consequences, taken in conjunction with other scientific principles, seem to be false” (PR, 65-66).

This takes place in the section where he also argues quite powerfully that "atomicity" and "continuity" are both consistent with universal relativity. So much for Zeno's paradox! I believe this is a definitive response. Without going into the significant differences between Einstein’s and Whitehead’s theories of relativity, they agree in principle that the simultaneity of actual entities is metaphysically baseless. If two entities did share the same ‘time-system’ then the universe would be deficient in novelty and fail to be creative—the ultimate category of Whitehead’s philosophy of organism. No entity’s world is simultaneous with another’s and that includes God on this account.

For simultaneous events in different spatial locations see other replies. But consider two colliding bodies: don't they reach the collision location simultaneously? The two events at the same location must happen simultaneously in order for the third event (collision) to come into existence.

Like Whitehead argues they would be "contemporaries" at best. To ignore this is to deny each entity the facticity of its own causal efficacy (transition) and presentational immediacies (spaceiness as concrescence). There is no need to posit the simultaneity unless you assume a block universe. . According to the Ontological Principle “actual entities are the only reasons; so that to search for a reason is to search for one or more actual entities. . According to the Ontological Principle “actual entities are the only reasons; so that to search for a reason is to search
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Myron Moses JacksonFeb 17 '13 at 17:39

for a reason is to search for one or more actual entities. It follows that any condition to be satisfied by one actual entity in its process expresses a fact either about the ‘real internal constitutions’ of some other actual entities, or about the ‘subjective aim’ conditioning that process” (PR, 24). To analyze events for Whitehead is committing a “fallacy of misplaced concreteness.” The actual occasion was somewhere—Whitehead doesn’t subscribe to a limited physicalism. His principle of relativity (not to be confused with mere applications of it like QM) holds that all that is real
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Myron Moses JacksonFeb 17 '13 at 17:40

and not simply actual (even unactualized possibility) exists as a densely web of relatedness to everything in the universe. Experienced a basketball game last night with thousands of people converged on same location. Was it simultaneity? Hardly. The game was just as real on a tv, computer, or radio. The fact that the game happened to occur at X does not become some metaphysical absolute. Alternative time systems are not only the source and reason for order in space but they are the meaning of possibility of motion and rest. Movement is stationary within a different time system,
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Myron Moses JacksonFeb 17 '13 at 17:41

which involves the co-gridients of “at the time sameness.” Whitehead is comfortable saying the percipients of the occasion were “practically simultaneous” (PR, 175) while denying simultaneities in the universe. Of course, he does not pretend that this working hypothesis to be “clear, obvious, and irreformable.”
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Myron Moses JacksonFeb 17 '13 at 17:41

but when the hand of a basketball player hits the ball, does the hand or the ball arrive to their meeting point first?
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artmFeb 17 '13 at 17:59